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Metrization criteria for compact groups in terms of their dense subgroups

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Dikranjan D., Shakhmatov D.

Fundamenta Mathematicae

2013

tom 221

nr 2

161-187

According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism Ĝ → D̂ of the dual groups is a topological isomorphism. We introduce four conditions on D that are necessary for it to determine G and we resolve the following question: If one of these conditions holds for every dense (or $G_δ$-dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined by all its $G_δ$-dense subgroups is metrizable, thereby resolving a question of Hernández, Macario and Trigos-Arrieta. (Under the additional assumption of the Continuum Hypothesis CH, the same statement was proved recently by Bruguera, Chasco, Domínguez, Tkachenko and Trigos-Arrieta.) As a tool, we develop a machinery for building $G_δ$-dense subgroups without uncountable compact subsets in compact groups of weight ω₁ (in ZFC). The construction is delicate, as these subgroups must have non-trivial convergent sequences in some models of ZFC.

Uniform entropy vs topological entropy

100%

Dikranjan D., Kunzi H.-P. A.

Topological Algebra and its Applications

2015

tom 3

nr 1

We discuss the connection between the topological entropy and the uniform entropy and answer several open questions from [10, 15]. We also correct several erroneous statements given in [10, 18] without proof.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Topological Algebra and its Applications

2 Dikranjan D.

1 Kunzi H.-P. A.

1 Shakhmatov D.

1 2015

1 2013

Wyniki wyszukiwania

Metrization criteria for compact groups in terms of their dense subgroups

Uniform entropy vs topological entropy